Average Error: 25.9 → 11.1
Time: 1.1m
Precision: 64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \mathsf{fma}\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}, \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\]
\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
↓
\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \mathsf{fma}\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}, \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)
double f(double d, double h, double l, double M, double D) {
        double r7221517 = d;
        double r7221518 = h;
        double r7221519 = r7221517 / r7221518;
        double r7221520 = 1.0;
        double r7221521 = 2.0;
        double r7221522 = r7221520 / r7221521;
        double r7221523 = pow(r7221519, r7221522);
        double r7221524 = l;
        double r7221525 = r7221517 / r7221524;
        double r7221526 = pow(r7221525, r7221522);
        double r7221527 = r7221523 * r7221526;
        double r7221528 = M;
        double r7221529 = D;
        double r7221530 = r7221528 * r7221529;
        double r7221531 = r7221521 * r7221517;
        double r7221532 = r7221530 / r7221531;
        double r7221533 = pow(r7221532, r7221521);
        double r7221534 = r7221522 * r7221533;
        double r7221535 = r7221518 / r7221524;
        double r7221536 = r7221534 * r7221535;
        double r7221537 = r7221520 - r7221536;
        double r7221538 = r7221527 * r7221537;
        return r7221538;
}
↓
double f(double d, double h, double l, double M, double D) {
        double r7221539 = d;
        double r7221540 = cbrt(r7221539);
        double r7221541 = h;
        double r7221542 = cbrt(r7221541);
        double r7221543 = r7221540 / r7221542;
        double r7221544 = sqrt(r7221543);
        double r7221545 = fabs(r7221543);
        double r7221546 = r7221544 * r7221545;
        double r7221547 = l;
        double r7221548 = cbrt(r7221547);
        double r7221549 = r7221540 / r7221548;
        double r7221550 = fabs(r7221549);
        double r7221551 = sqrt(r7221549);
        double r7221552 = r7221550 * r7221551;
        double r7221553 = M;
        double r7221554 = D;
        double r7221555 = r7221553 * r7221554;
        double r7221556 = 2.0;
        double r7221557 = r7221556 * r7221539;
        double r7221558 = r7221555 / r7221557;
        double r7221559 = r7221541 * r7221558;
        double r7221560 = r7221558 * r7221559;
        double r7221561 = -0.5;
        double r7221562 = r7221561 / r7221547;
        double r7221563 = r7221560 * r7221562;
        double r7221564 = fma(r7221552, r7221563, r7221552);
        double r7221565 = r7221546 * r7221564;
        return r7221565;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

  1. Initial program 25.9

    \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

  2. Simplified26.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}\right) \cdot \frac{h \cdot \frac{-1}{2}}{\ell}, \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{h}}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt26.5

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}\right) \cdot \frac{h \cdot \frac{-1}{2}}{\ell}, \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}}\]

  5. Applied add-cube-cbrt26.6

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}\right) \cdot \frac{h \cdot \frac{-1}{2}}{\ell}, \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}\]

  6. Applied times-frac26.6

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}\right) \cdot \frac{h \cdot \frac{-1}{2}}{\ell}, \sqrt{\frac{d}{\ell}}\right) \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}}}\]

  7. Applied sqrt-prod21.0

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}\right) \cdot \frac{h \cdot \frac{-1}{2}}{\ell}, \sqrt{\frac{d}{\ell}}\right) \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)}\]

  8. Simplified20.2

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}\right) \cdot \frac{h \cdot \frac{-1}{2}}{\ell}, \sqrt{\frac{d}{\ell}}\right) \cdot \left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]
  9. Using strategy rm

  10. Applied *-un-lft-identity20.2

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}\right) \cdot \frac{h \cdot \frac{-1}{2}}{\color{blue}{1 \cdot \ell}}, \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  11. Applied times-frac20.2

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}\right) \cdot \color{blue}{\left(\frac{h}{1} \cdot \frac{\frac{-1}{2}}{\ell}\right)}, \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  12. Applied associate-*r*19.1

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \color{blue}{\left(\left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}\right) \cdot \frac{h}{1}\right) \cdot \frac{\frac{-1}{2}}{\ell}}, \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  13. Simplified17.7

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \color{blue}{\left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right)} \cdot \frac{\frac{-1}{2}}{\ell}, \sqrt{\frac{d}{\ell}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]
  14. Using strategy rm

  15. Applied add-cube-cbrt17.8

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \sqrt{\frac{d}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  16. Applied add-cube-cbrt17.9

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \sqrt{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  17. Applied times-frac17.9

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \sqrt{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  18. Applied sqrt-prod15.7

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \color{blue}{\sqrt{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  19. Simplified15.4

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\ell}}, \left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]
  20. Using strategy rm

  21. Applied add-cube-cbrt15.4

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{d}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}, \left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  22. Applied add-cube-cbrt15.5

    \[\leadsto \mathsf{fma}\left(\sqrt{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}, \left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  23. Applied times-frac15.5

    \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}, \left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  24. Applied sqrt-prod11.6

    \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}, \left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  25. Simplified11.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}, \left(\frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot h\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

  26. Final simplification11.1

    \[\leadsto \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \mathsf{fma}\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}, \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(h \cdot \frac{M \cdot D}{2 \cdot d}\right)\right) \cdot \frac{\frac{-1}{2}}{\ell}, \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))